LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

For a member A of a lambda-system D, the sets B with A intersection B in D form a lambda-system

Statement

Let $D$ be a lambda-system on $X$ and fix $A \in D$ . Then

$D_{A} : = {B \in D : A \cap B \in D}$

is a lambda-system on $X$ .

Facts & Assumptions

Given: A lambda-system $D$ on $X$ and a fixed set $A \in D$ .

[L1]

A lambda-system contains $X$ , is closed under relative differences, and is closed under increasing countable unions (Lambda-systems, or Dynkin systems).

Proof

technique · direct

1.1givenL1

One has $X \in D$ and $A \cap X = A \in D$ , so $X \in D_{A}$ .

1.2L1algebra

If $B, C \in D_{A}$ and $B \subseteq C$ , then $C ∖ B \in D$ and $A \cap (C ∖ B) = (A \cap C) ∖ (A \cap B) \in D$ by [L1]; hence $C ∖ B \in D_{A}$ .

2.1L1algebra∎

If $(B_{n})$ is increasing in $D_{A}$ , then $⋃_{n} B_{n} \in D$ and $A \cap ⋃_{n} B_{n} = ⋃_{n} (A \cap B_{n}) \in D$ by [L1]. Therefore $⋃_{n} B_{n} \in D_{A}$ , and all lambda-system axioms hold.

Depends on

Lambda-systems, or Dynkin systems

Used by

The lambda-system generated by a pi-system is closed under finite intersections Lemma

Dependency tree · one level

1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Dembo, Probability Theory lecture notes, proof of Theorem 1.1.38 (standard reference, not scraped)